A couple of months ago I stumbled across Paul Lockhart's essay A Mathematician's Lament and it made perfect sense to me. I'm not meaning to argue this essay one way or the other, except to say that 12 years of what I did in math class really isn't mathematics as you -- and I, as an "enthusiastic amateur" -- enjoy it.

We're probably going to homeschool our daughter, who will be kindergarten age next fall. I feel there's a place for knowing your times tables and the like, but there's also a place for knowing that mathematics is more than arithmetic and formulas: it's discovery, playing with ideas, etc.

Where I'm going is that I know enough to understand the difference, but I'm not quite confident enough to teach (or lead) this process effectively, since I'm not a professional mathematician.

Are there any curricula that would help provide some structure to facilitate this kind of learning, so that there is one less student who has a shortchanged opinion of the mathematics profession?

Or, alternatively, if you felt that your elementary school math education hit the mark, how was it done?

I converted the question to wiki for you. The "community wiki" checkbox should be just under the input field, above the preview.
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Anton GeraschenkoNov 11 '09 at 18:07

I thought sure I looked for it there ... I'm not a stranger to Stack* sites. But thanks for fixing it in any case.
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user1020Nov 11 '09 at 19:28

The CW checkbox will not appear until you have amassed at least 10 reputation points so it may not have been there before asking the question.
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JasonJan 7 '11 at 1:06

Just to mention this, I think it would be an excellent idea for the mathematics community to "crowdsource" a pool of inquiry-based questions that develop the ideas of the elementary mathematics curriculum in interesting ways. Such a list of questions could be compiled into a good IBL script for teaching mathematics "from the ground up".
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Jon BannonOct 8 '13 at 23:18

"Are there elementary-school curricula that capture the joy of mathematics?" or even "Are there elementary-school curricula that capture mathematics?". I'd answer but I have learned better than that :-)
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Włodzimierz HolsztyńskiOct 9 '13 at 3:17

11 Answers
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As far as a full curriculum goes, I don't believe there is one that does exactly what you want. Books (in the United States, at least) divide into two camps:

"Constructivist" (e.g. Everyday Math, Connected Math)

"Traditional" (e.g. Saxon, Singapore)

Now, any search you make that even has a whiff of these terms will summon up loud and angry missives (try this article from the New York Times for an idea).

Constructivist curriculum is an attempt to catch the "joy of mathematics" approach to learning; for example rather than a worksheet with addition problems there might be a question about all the different possible sets of numbers that add up to 20.

The downside (as pointed out by the article above) is that (especially when taught by teachers who aren't themselves strong in mathematics) it can lead to basic skills being missed.

This is a problem Lockheart's Lament acknowledges. He seems to think students won't miss anything important. This can be true if the person steering the education is a mathematician, but with a non-specialist (i.e. most elementary school educators and homeschoolers) things can go horribly wrong.

Now, it's possible to balance to pull off a fantastic curriculum, but the ones I know about (say, at the Russian School of Mathematics in Boston) are, as self-described by the teachers, not following a curriculum at all. That's great if the teachers are experts, but put homeschoolers in a quandry.

I think the world is still waiting for an inquiry-type elementary curriculum that can be followed by non-experts and doesn't shortchange basic skills. So for now I'd suggest:

a.) Pick a traditional curriculum (Singapore is fine, although do shop around).

b.) Supplement. This very question is filling with lots of suggestions.

I would read Papert's Mindstorms and "The Children's Machine", as they better explain the constructivist/constructionist point of view. A textbook is the antithesis of that philosophy, so I have to agree that if you are homeschooling and are not an expert, you should use a traditional curriculum. However, you might read what people like Paolo Blikstein, Seymour Papert, and Idit Harel have done with computers in teaching mathematics.
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Michael HoffmanNov 12 '09 at 0:39

Our kids get their math at home; a combination of Singapore math and Gelfand's algebra.
The second ref is the "beuty of math" stuff you refer to; obligatory quote: "Problem 231: Find a recording of Bach's Well-Tempered Clavier and enjoy it". Problem level is certainly higher than Kindergarden though. Gelfand wrote some other books aimed at kids, but IMHO the other ones are not that nice.

Hello!
I am mathematician and a homeschooler (that is - my children are homeschooled).
If you want to capture the joy of mathematics, or the joy of anything for that matter, then it is my personal belief that the best curriculum is no curriculum. That last sentence is a bit of an exaggeration, but still my experience has shown that the most important things children discover themselves, and the most joyful moments of discovery are when one discovers something for oneself. Since you emphasize joy in your question, this is my answer.

Er... I hope you don't mean this literally. Truly non-structured education (where you don't think about what to cover in what order, what is a prerequisite to what, etc.) seems to me like it would always be truly horrible (at least, it always was for me). An approach with "no curriculum" would only seem appropriate to me if you are so comfortable teaching mathematics that you know a curriculum innately, i.e. if you already know innately which concepts you want your children to discover, in what order, and how to make sure they do.
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Ilya GrigorievDec 18 '09 at 17:24

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I am a mathematician who was homeschooled, and is now homeschooling his children. In high school I was "unschooled" as Orr describes. My parents were neither educators, nor did they know what materials were best, and so I just bought a bunch of books about what I was interested in. The books necessarily organized the information of their respective topics. I think that any "loss" in this approach is offset by the deep engagement that pure inquiry allows. At the end of the day, I'm not a Fields medalist but still love mathematics and so my work is a natural extension of what I have always done.
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Jon BannonOct 7 '13 at 14:27

Having written the above, though, let me say that I plan on providing a Dewey-esque "organization of experience" for my children. I plan to attempt to unobtrusively organize what they encounter as Ilya describes. Provided this can be done well, it will maintain the spirit of inquiry and avoid a lot of wheel-spinning.
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Jon BannonOct 7 '13 at 14:32

And there are schools that lack curricula. In "Sudbury" schools, i.e. those that emulate the Sudbury Valley School, no one is given any disciplined instruction in any subject until and unless they request it. The original SVS claims that in more than four decades of their existence, all of their pupils have learned to read before leaving and 80% have gone on to colleges or universities. Schools with curricula can't generally claim those things.

I think the requirement that all pupils must learn mathematics has led to a lot of dishonesty: University graduates have been taught that mathematics consists of clerical skills to be memorized and followed algorithmically.

The Montessori materials for teaching arithmetic and even algebra (the exact same materials work for both!) are among the best I've ever seen. They are physical and concrete and yet convey correctly the abstract concepts involved.
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Deane YangJan 6 '11 at 17:52

Arthur Benjamin has a series of video lectures called The Joy of Mathematics which I think is pretty good. It's not a curriculum, but all the lectures (or at least I'll the ones I've seen) introduce some new concepts and explore them in a pretty nice way. I'm not sure if it's appropriate kindergarten material, but I'm pretty sure most of it is appropriate elementary school material. Here's a list of the lecture titles:

The Joy of Math—The Big Picture

The Joy of Numbers

The Joy of Primes

The Joy of Counting

The Joy of Fibonacci Numbers

The Joy of Algebra

The Joy of Higher Algebra

The Joy of Algebra Made Visual

The Joy of 9

The Joy of Proofs

The Joy of Geometry

The Joy of Pi

The Joy of Trigonometry

The Joy of the Imaginary Number i

The Joy of the Number e

The Joy of Infinity

The Joy of Infinite Series

The Joy of Differential Calculus

The Joy of Approximating with Calculus

The Joy of Integral Calculus

The Joy of Pascal's Triangle

The Joy of Probability

The Joy of Mathematical Games

The Joy of Mathematical Magic

Benjamin's book Secrets of Mental Math also looks good for teaching kids to play around with arithmetic and develop number sense.

Personally I was not homeschooled, but from around 4th grade I attended a math circle in Boston. Many of the other students were homeschooled, and I think this sort of thing fills exactly the niche you are asking about. So I suggest looking into math circles in your area, although with the caveat that depending on where you are, the local math circles may be targets at older students than the one in Boston (which is particularly welcoming for kindergarten-aged kids) and/or more focussed on problem-solving. The Boston math circle is collaborative and inquiry based. As an example, a class of kindergarten kids might spend 10 weeks playing with math, starting with a question like "Are there numbers between numbers?" or "How many squares fit in a circle?"

Another recommendation: Conway's "The book of numbers" and Courant & Robbins' "What is Mathematics?" both have a wealth of material of sort you're interested in, perfectly suited to elementary school students (although perhaps only after they have some foundation in basic arithmetic).

And maybe Sam feels guilty plugging a book for which he provided the appendix, but "Out of the Labyrinth" is a really fantastic read for anyone who wants to know more about the Math Circle.
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Jonah OstroffNov 12 '09 at 4:47

In our kitchen, as the boys were growing up, we had "count-by" tables posted at random locations. In order to teach my youngest son how to multiply two digit numbers, we focused upon learning squares. This idea proceeded along the lines of
$20\times 20 =400$. What is $2\times 20$? What is $2\times 20 +1$? What is $20\times20 + 2 \times 20 +1$? What is $21\times 21$? Essentially, I taught him to internalize $n^2+2n+1$, $n^2-2n+1$, $n^2 \pm 4n+4$, and $n^2\pm 6n +9$. In this way, if a square was proximate to a square he knew, he could compute the unknown square. Approximating products as the square of the average is a good trick, and finally, you can compute the product of two numbers of the same parity by squaring the average and subtracting the square of the difference to the average. If the numbers have differing parity, then you need to subtract numbers of the form $n(n+1)$. A lot of algebraic rules become easier to understand if alternative algorithms to multiplication are employed.

I also strongly encourage counting by eggs (via dozens) $1/12$, $1/6$, $1/4$, $1/3$, $5/12$, $1/2$, $7/12$, $2/3$, $3/4$, $5/6$, $11/12$, 1 dozen. And counting by other fractional quantities. You will have to train yourself to do the mental arithmetic as you teach your children.

Finally, look at puzzle books and problem solving books, and look for games that involve dice and (ordinary) playing cards.

There are many entry points into a love of mathematics. If by elementary school you mean grades k-5 there are not a lot of tools that such students have so pattern hunting is part of the fun. However, in a general way, I think geometry and combinatorial geometry have been short changed as an entry, especially for very young kids. As students get further along in their education I think the possibilities pick up.

In my own opinion I know of no better way to get started attracting kids to mathematics than the books of Martin Gardner:

However, many children of parents who love mathematics may not share this passion with their parents.

I also regret that knowing the role of mathematics in new technologies and the applications of mathematics in general get short shrift with people of all ages. I am biased because I am one of the co-author of this book, but I think the book For All Practical Purposes (FAPP) gives some of the sense of excitement involved in using mathematics in a variety of settings. FAPP has been through many editions and some of the early editions are probably available cheaply on the web.
I was involved with the first 4 chapters which deal with urban operations research problems (Eulerian circuits- applications to street sweeping, etc.; Hamiltonian circuits - applications to package pick up and delivery, etc.) This book is designed for liberal arts students in college but parents may find it of interest, too.

Love Martin Gardner! Also appreciate you comment about the possibility that my daughter won't love mathematics as much as I. But at the very least I want her to know that mathematics isn't just what is in her school math book.
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user1020Nov 11 '09 at 19:39

There is indeed a lot of mathematics that does not appear in any particular book. However, since your daughter is still so very young (not yet in kindergarten) you might consider getting her "mathematical toys" which you can have fun playing with along with her. For example, d-stix, jovo toys, polydrons (though some of these have small parts which may be dangerous for small children) make it possible to construct various kinds of polyhedra. There are also kits for making tiling patterns. It seems to me some of the comments made here apply primarily to older children.
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Joseph MalkevitchNov 12 '09 at 4:36

Here is something I've considered. I only have the first volume, though, and am not sure if it gets at the mathematics the way I'd like. One thing I can say is that the book is very funny to my 5 year old.

Another resource that I've heard is good but have not yet tried is Beast Academy, which was produced by the Art of Problem Solving. These present mathematics topics in a graphic novel format. Children purportedly have a hard time putting them down.